II. OPEN
CHANNEL FLOWS THEORY

Specific energy (specific
charge) in a cross section of a channel or a river is:

the average value of the energy
of the molecules of the liquid of this section, by unit of weight of this
liquid, compared to the

* horizontal
one passing by the point low of this section; * the average
load of the section compared to a privileged datum-line; * the distance enters
the line of load and the bottom a given section.

From Bernoulli's equation,
the total energy per unit mass H in a turbulent open channel flow is given
by

where y is the flow depth, z
the bed height above datum, g the acceleration due to gravity, and V the
average velocity. In this equation, the velocity coefficient associated
with the term V2/2g is set to unity, as usual for uniform flow.
Without losses, H = constant, and flow over an elevated section of bed
of height D z is described by

where thespecific
energy E is given by

in which subscripts 1 and 2 refer
to locations upstream and downstream of the change in bed elevation, and
q is the flow rate per metre width of channel (Figure 1).

Figure 1

Given upstream conditions, the value
of E downstream is obtained, and consequently the downstream flow depth.
For a particular vale of E, there are two possible values of y, as shown
in Figure 2.

II.2 Froude
number :

The appropriate value of y is determined by the upstream
value of the
Froude number

Figure 2

For Fr > 1, the flow
is supercritical and is fast and shallow (the lower arm of Figure 2), and
for Fr < 1, the flow is subcritical and is slow and deep. Thus if the
upstream flow is subcritical (E1, y1 in Figure 2),
flow over the obstacle will be subcritical at reduced E, corresponding
to reduced y (E2, y2 in Figure 2); the flow decreases
over the raised bed. If
D z is sufficiently large to attempt to force E2
to become less than the minimum possible value, the upstream flow will
adjust, increasing the upstream flow depth so that E2 is the
minimum value. In this case the structure is a control, and the flow depth
yc, velocity Vc, and flowrate q at the minimum energy
(critical) point are related by

and.

If the bed level subsequently
decreases, the critical point must occur at the maximum elevation, beyond
which the flow becomes upercritical. A decrease in bed level corresponds
to an increase in E, and consequently a decrease in y as the E, y relationship
moves along the lower arm of Figure 2. Further downstream the imposed head
may force a flow depth to be above that possible for a supercritical flow,
in which case theflow must change back to subcritical. This is done by
means of a hydraulic jump.

Thus for a given q, in
the absence of losses the flowing depth over the maximum elevation is given
by the critical condition. Conversely, a measurement of the flow depth
at the critical point, which must be the crest, provides a measure of q.
Downstream of the critical point, the flow is supercritical and is controlled
only by the structure itself. If the downstream head condition is sufficiently
high, the downstream jump may drown the obstacle, in which case the flow
remains subcritical throughout.